No. If this debt is primarily consumer in nature, the the Fair Debt Collection Practices Act, both federal and California, should apply and prohibit such harassing calls to others that reveal embarrassing information to third parties. Was this a loan with a payday lender? They are usually the worst and often do not worry about the FDCPA, they just go on their merry way harassing people and risk getting sued for the consumer's actual damages and attorney's fees.
You should immediately consult a consumer attorney (see link below) who handles debt collection harassment, to see if this is worth pursuing, as not all debt collectors are even worth suing, as they work out of some weird place, such as an apartment, with many phones, calling and harassing until they get shut down by the Attorney General, the FTC, or sued out of existence.
Robert Stempler (please see DISCLAIMER below)
NOTICE: The above statements are provided for general information purposes only and are not intended as legal advice or advice of any sort for a specific case or legal matter. If you do not have a signed attorney-client fee agreement with the Consumer Law Office of Robert Stempler, APC ("the Firm"), then until such written fee agreement is provided and signed by both a prospective client and attorney for a particular case, neither Mr. Stempler nor the Firm will represent you nor will they be your attorney in any matter and you remain responsible for retaining your own attorney and for compliance with any and all deadlines and for any statutes of limitations that may pertain to potential claims. Comments made on a public forum, such as Avvo.com, to not have any confidentiality because others may read them. If you desire a private consultation with Mr. Stempler that is confidential, please go to www.StopCollectionLawsuits.com and submit a free eCase Review. The result portrayed for a client was dependent on the facts of that case. Results will differ if based on different facts. The Firm and Mr. Stempler are a debt relief agency. The Firm and Mr. Stempler help people file for bankruptcy relief under the Bankruptcy Code.
If your credit card debt was not for a business, the collection agency is violating the Fair Debt Collection Practices Act (FDCPA). They are allowed to call third parties to locate you, but once the agency knows your location information, they cannot call third parties. The agency cannot disclose the existence of the debt to any third parties (except your spouse). They can come to your workplace to serve with a law suit, but it sounds like the threat is more to intimidate you. Under the FDCPA, if an agency has violated the law, you are entitled to up to $1000, and they must pay your reasonable attorneys fees and costs.